Preheat your oven to 375º. Prepare a baking sheet by basting it with a light layer of safflower oil. About 1 tbsp.
In a large bowl, combine the scallion whites, garlic, ginger, cubed tofu, shiitake mushrooms, carrot and teriyaki sauce. Toss to coat.
Spread the lettuce wrap filling on the baking sheet and bake on the middle rack for 25-30 minutes until the mixture has a light crispy texture.
While the filling is baking, separate the leaves from the head of butterhead lettuce. Rinse and dry thoroughly and set aside.
To make the dipping sauce, combine the minced garlic and ginger with the lime juice, sugar, tamari, safflower oil and crushed red pepper flakes into a small food processor. Blend until well combined.
To serve, put the tofu filling in a bowl with the lettuce leaves, cabbage, red pepper, limes, scallion greens, sesame seeds and Asian Dipping Sauce on the side. Make each wrap by spooning the filling into the lettuce leaf and topping as desired.